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Rewrite the equation 4x3x−12+3=10

in two equations joined by “and.”

11 answers

You can rewrite the given equation \( 4x^3 - 12 + 3 = 10 \) as two equations joined by "and" as follows:

\( 4x^3 - 12 + 3 = 10 \) and \( 4x^3 - 12 + 3 - 10 = 0 \).

This simplifies to:

\( 4x^3 - 9 = 10 \) and \( 4x^3 - 9 = 0 \).

Can you answer it step by step

Certainly! Let's break down the process of rewriting the equation \( 4x^3 - 12 + 3 = 10 \) into two equations joined by "and" step by step.

Step 1: Understand the Original Equation

The original equation is: \[ 4x^3 - 12 + 3 = 10 \]

Step 2: Simplify the Left Side

First, simplify the left side of the equation:

  • Combine the constants \(-12\) and \(+3\): \[ -12 + 3 = -9 \] So the equation becomes: \[ 4x^3 - 9 = 10 \]

Step 3: Rearrange the Equation

Now, we can rearrange this equation to isolate one side: \[ 4x^3 - 9 - 10 = 0 \] This simplifies to: \[ 4x^3 - 19 = 0 \]

Step 4: Write Two Equations

We now have two forms:

  1. The simplified version of the original equation: \[ 4x^3 - 9 = 10 \]
  2. The arrangement set to zero: \[ 4x^3 - 19 = 0 \]

Step 5: Combine with "and"

We can now write these as two equations joined by "and": \[ 4x^3 - 9 = 10 \quad \text{and} \quad 4x^3 - 19 = 0 \]

Final Result

Thus, the original equation can be expressed as: \[ 4x^3 - 9 = 10 \quad \text{and} \quad 4x^3 - 19 = 0 \]

This concludes the step-by-step process!

can you answer it simply, but also step by step

Of course! Here's a simplified step-by-step breakdown of rewriting the equation \( 4x^3 - 12 + 3 = 10 \) into two equations joined by "and."

Step 1: Start with the Original Equation

The original equation is: \[ 4x^3 - 12 + 3 = 10 \]

Step 2: Simplify the Left Side

Combine the constants on the left side: \[ -12 + 3 = -9 \] So, we rewrite the equation as: \[ 4x^3 - 9 = 10 \]

Step 3: Rearrange to Set One Side to Zero

Now, we manipulate the equation to set one side to zero: \[ 4x^3 - 9 - 10 = 0 \] This simplifies to: \[ 4x^3 - 19 = 0 \]

Step 4: Write Two Equations

Now we can write the two equations:

  1. From the original format: \[ 4x^3 - 9 = 10 \]
  2. From setting it to zero: \[ 4x^3 - 19 = 0 \]

Step 5: Combine with "and"

We can now express these as: \[ 4x^3 - 9 = 10 \quad \text{and} \quad 4x^3 - 19 = 0 \]

Final Result

The final answer is: \[ 4x^3 - 9 = 10 \quad \text{and} \quad 4x^3 - 19 = 0 \]

This completes the process!

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Demonstrate how to solve the equation 3.51.2x=5.4
for the variable x. Make sure to show each step.

Sure! Let's solve the equation \( 3.5 \cdot 1.2^x = 5.4 \) for the variable \( x \) step by step.

Step 1: Isolate the Exponential Term

We start with the equation: \[ 3.5 \cdot 1.2^x = 5.4 \] To isolate \( 1.2^x \), divide both sides of the equation by \( 3.5 \): \[ 1.2^x = \frac{5.4}{3.5} \]

Step 2: Calculate the Right Side

Now, calculate \( \frac{5.4}{3.5} \): \[ \frac{5.4}{3.5} \approx 1.542857 \] So we have: \[ 1.2^x \approx 1.542857 \]

Step 3: Take the Logarithm of Both Sides

To solve for \( x \), take the logarithm (we can use natural logarithm \( \ln \) or common logarithm \( \log \)). Here we will use natural logarithm: \[ \ln(1.2^x) = \ln(1.542857) \]

Step 4: Use the Power Rule of Logarithms

By applying the power rule of logarithms \( \ln(a^b) = b \cdot \ln(a) \), we get: \[ x \cdot \ln(1.2) = \ln(1.542857) \]

Step 5: Solve for \( x \)

Now, divide both sides by \( \ln(1.2) \) to solve for \( x \): \[ x = \frac{\ln(1.542857)}{\ln(1.2)} \]

Step 6: Calculate the Values

Now, use a calculator to find the values of the logarithms:

  • \( \ln(1.542857) \approx 0.4343 \)
  • \( \ln(1.2) \approx 0.1823 \)

Substituting these values into the equation: \[ x \approx \frac{0.4343}{0.1823} \approx 2.384 \]

Final Answer

Thus, the solution for \( x \) is approximately: \[ x \approx 2.384 \]

If you need further assistance or clarification, feel free to ask!

Can you simplify the steps and make it more straightforward and broad?

Absolutely! Here’s a simplified and more straightforward approach to solving the equation \( 3.5 \cdot 1.2^x = 5.4 \):

Step 1: Isolate the Exponential Term

  • Start with the equation: \[ 3.5 \cdot 1.2^x = 5.4 \]
  • Divide both sides by \( 3.5 \): \[ 1.2^x = \frac{5.4}{3.5} \]

Step 2: Simplify the Right Side

  • Calculate \( \frac{5.4}{3.5} \): \[ 1.2^x = 1.542857 \quad (\text{approximately}) \]

Step 3: Take the Logarithm

  • Apply logarithm to both sides (we can use common logarithm \( \log \) or natural logarithm \( \ln \)): \[ \log(1.2^x) = \log(1.542857) \]

Step 4: Use the Power Rule

  • Using the power rule \( \log(a^b) = b \cdot \log(a) \): \[ x \cdot \log(1.2) = \log(1.542857) \]

Step 5: Solve for \( x \)

  • Rearrange to solve for \( x \): \[ x = \frac{\log(1.542857)}{\log(1.2)} \]

Step 6: Calculate the Values

  • Use a calculator to find the logarithmic values and compute \( x \):
    • \( \log(1.542857) \approx 0.187 \)
    • \( \log(1.2) \approx 0.079 \)

Using these values: \[ x \approx \frac{0.187}{0.079} \approx 2.37 \quad (\text{approximately}) \]

Final Answer

Thus, the solution is: \[ x \approx 2.37 \]

This version condenses the steps while providing a clear direction for solving the equation! If you have any further questions, feel free to ask.